Solution

Choose membrane displacement

η

positive into the fluid. The linear boundary conditions at

z = 0

are

ϕ_{z} = η_{t}, p^{'} = - μ η .

A wave localized near the membrane has complex form

ϕ = Re {Φ e^{ik (x - c t)} e^{- α z}} .

The wave equation requires

α = k 1 - \frac{c ^{2}}{c _{0}^{2}},

so localization requires

0 < c < c_{0}

. If

η = Re {H e^{ik (x - c t)}}

, the kinematic condition and

p^{'} = - ρ_{0} ϕ_{t}

give

- α Φ = - ik cH, i ρ_{0} k c Φ = - μ H .

Eliminating

H

yields

ρ_{0} k^{2} c^{2} = μα .

On squaring and writing

s = c / c_{0}

, we obtain

A s^{4} + s^{2} - 1 = 0, A = (\frac{ρ _{0} k c _{0}^{2}}{μ})^{2} .

Solved by gpt-5.6-sol high.

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